Properties

Label 483.2.a.b.1.1
Level $483$
Weight $2$
Character 483.1
Self dual yes
Analytic conductor $3.857$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 483.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +4.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +4.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{9} +8.00000 q^{10} -5.00000 q^{11} +2.00000 q^{12} -2.00000 q^{13} -2.00000 q^{14} +4.00000 q^{15} -4.00000 q^{16} +2.00000 q^{18} -5.00000 q^{19} +8.00000 q^{20} -1.00000 q^{21} -10.0000 q^{22} -1.00000 q^{23} +11.0000 q^{25} -4.00000 q^{26} +1.00000 q^{27} -2.00000 q^{28} -2.00000 q^{29} +8.00000 q^{30} +6.00000 q^{31} -8.00000 q^{32} -5.00000 q^{33} -4.00000 q^{35} +2.00000 q^{36} +6.00000 q^{37} -10.0000 q^{38} -2.00000 q^{39} +5.00000 q^{41} -2.00000 q^{42} +8.00000 q^{43} -10.0000 q^{44} +4.00000 q^{45} -2.00000 q^{46} -9.00000 q^{47} -4.00000 q^{48} +1.00000 q^{49} +22.0000 q^{50} -4.00000 q^{52} +9.00000 q^{53} +2.00000 q^{54} -20.0000 q^{55} -5.00000 q^{57} -4.00000 q^{58} +9.00000 q^{59} +8.00000 q^{60} -5.00000 q^{61} +12.0000 q^{62} -1.00000 q^{63} -8.00000 q^{64} -8.00000 q^{65} -10.0000 q^{66} +4.00000 q^{67} -1.00000 q^{69} -8.00000 q^{70} +12.0000 q^{71} +12.0000 q^{74} +11.0000 q^{75} -10.0000 q^{76} +5.00000 q^{77} -4.00000 q^{78} -10.0000 q^{79} -16.0000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -18.0000 q^{83} -2.00000 q^{84} +16.0000 q^{86} -2.00000 q^{87} +10.0000 q^{89} +8.00000 q^{90} +2.00000 q^{91} -2.00000 q^{92} +6.00000 q^{93} -18.0000 q^{94} -20.0000 q^{95} -8.00000 q^{96} -18.0000 q^{97} +2.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.00000 1.00000
\(5\) 4.00000 1.78885 0.894427 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 2.00000 0.816497
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 8.00000 2.52982
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 2.00000 0.577350
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −2.00000 −0.534522
\(15\) 4.00000 1.03280
\(16\) −4.00000 −1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 2.00000 0.471405
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 8.00000 1.78885
\(21\) −1.00000 −0.218218
\(22\) −10.0000 −2.13201
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) −4.00000 −0.784465
\(27\) 1.00000 0.192450
\(28\) −2.00000 −0.377964
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 8.00000 1.46059
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) −8.00000 −1.41421
\(33\) −5.00000 −0.870388
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) 2.00000 0.333333
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −10.0000 −1.62221
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 5.00000 0.780869 0.390434 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) −2.00000 −0.308607
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −10.0000 −1.50756
\(45\) 4.00000 0.596285
\(46\) −2.00000 −0.294884
\(47\) −9.00000 −1.31278 −0.656392 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) −4.00000 −0.577350
\(49\) 1.00000 0.142857
\(50\) 22.0000 3.11127
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 2.00000 0.272166
\(55\) −20.0000 −2.69680
\(56\) 0 0
\(57\) −5.00000 −0.662266
\(58\) −4.00000 −0.525226
\(59\) 9.00000 1.17170 0.585850 0.810419i \(-0.300761\pi\)
0.585850 + 0.810419i \(0.300761\pi\)
\(60\) 8.00000 1.03280
\(61\) −5.00000 −0.640184 −0.320092 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) 12.0000 1.52400
\(63\) −1.00000 −0.125988
\(64\) −8.00000 −1.00000
\(65\) −8.00000 −0.992278
\(66\) −10.0000 −1.23091
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) −8.00000 −0.956183
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 12.0000 1.39497
\(75\) 11.0000 1.27017
\(76\) −10.0000 −1.14708
\(77\) 5.00000 0.569803
\(78\) −4.00000 −0.452911
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) −16.0000 −1.78885
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) −18.0000 −1.97576 −0.987878 0.155230i \(-0.950388\pi\)
−0.987878 + 0.155230i \(0.950388\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 16.0000 1.72532
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 8.00000 0.843274
\(91\) 2.00000 0.209657
\(92\) −2.00000 −0.208514
\(93\) 6.00000 0.622171
\(94\) −18.0000 −1.85656
\(95\) −20.0000 −2.05196
\(96\) −8.00000 −0.816497
\(97\) −18.0000 −1.82762 −0.913812 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 2.00000 0.202031
\(99\) −5.00000 −0.502519
\(100\) 22.0000 2.20000
\(101\) 5.00000 0.497519 0.248759 0.968565i \(-0.419977\pi\)
0.248759 + 0.968565i \(0.419977\pi\)
\(102\) 0 0
\(103\) −19.0000 −1.87213 −0.936063 0.351833i \(-0.885559\pi\)
−0.936063 + 0.351833i \(0.885559\pi\)
\(104\) 0 0
\(105\) −4.00000 −0.390360
\(106\) 18.0000 1.74831
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 2.00000 0.192450
\(109\) −12.0000 −1.14939 −0.574696 0.818367i \(-0.694880\pi\)
−0.574696 + 0.818367i \(0.694880\pi\)
\(110\) −40.0000 −3.81385
\(111\) 6.00000 0.569495
\(112\) 4.00000 0.377964
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) −10.0000 −0.936586
\(115\) −4.00000 −0.373002
\(116\) −4.00000 −0.371391
\(117\) −2.00000 −0.184900
\(118\) 18.0000 1.65703
\(119\) 0 0
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) −10.0000 −0.905357
\(123\) 5.00000 0.450835
\(124\) 12.0000 1.07763
\(125\) 24.0000 2.14663
\(126\) −2.00000 −0.178174
\(127\) 9.00000 0.798621 0.399310 0.916816i \(-0.369250\pi\)
0.399310 + 0.916816i \(0.369250\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) −16.0000 −1.40329
\(131\) −5.00000 −0.436852 −0.218426 0.975854i \(-0.570092\pi\)
−0.218426 + 0.975854i \(0.570092\pi\)
\(132\) −10.0000 −0.870388
\(133\) 5.00000 0.433555
\(134\) 8.00000 0.691095
\(135\) 4.00000 0.344265
\(136\) 0 0
\(137\) 9.00000 0.768922 0.384461 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) −2.00000 −0.170251
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) −8.00000 −0.676123
\(141\) −9.00000 −0.757937
\(142\) 24.0000 2.01404
\(143\) 10.0000 0.836242
\(144\) −4.00000 −0.333333
\(145\) −8.00000 −0.664364
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 12.0000 0.986394
\(149\) 3.00000 0.245770 0.122885 0.992421i \(-0.460785\pi\)
0.122885 + 0.992421i \(0.460785\pi\)
\(150\) 22.0000 1.79629
\(151\) −19.0000 −1.54620 −0.773099 0.634285i \(-0.781294\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 10.0000 0.805823
\(155\) 24.0000 1.92773
\(156\) −4.00000 −0.320256
\(157\) 7.00000 0.558661 0.279330 0.960195i \(-0.409888\pi\)
0.279330 + 0.960195i \(0.409888\pi\)
\(158\) −20.0000 −1.59111
\(159\) 9.00000 0.713746
\(160\) −32.0000 −2.52982
\(161\) 1.00000 0.0788110
\(162\) 2.00000 0.157135
\(163\) 13.0000 1.01824 0.509119 0.860696i \(-0.329971\pi\)
0.509119 + 0.860696i \(0.329971\pi\)
\(164\) 10.0000 0.780869
\(165\) −20.0000 −1.55700
\(166\) −36.0000 −2.79414
\(167\) 19.0000 1.47026 0.735132 0.677924i \(-0.237120\pi\)
0.735132 + 0.677924i \(0.237120\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −5.00000 −0.382360
\(172\) 16.0000 1.21999
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) −4.00000 −0.303239
\(175\) −11.0000 −0.831522
\(176\) 20.0000 1.50756
\(177\) 9.00000 0.676481
\(178\) 20.0000 1.49906
\(179\) 8.00000 0.597948 0.298974 0.954261i \(-0.403356\pi\)
0.298974 + 0.954261i \(0.403356\pi\)
\(180\) 8.00000 0.596285
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 4.00000 0.296500
\(183\) −5.00000 −0.369611
\(184\) 0 0
\(185\) 24.0000 1.76452
\(186\) 12.0000 0.879883
\(187\) 0 0
\(188\) −18.0000 −1.31278
\(189\) −1.00000 −0.0727393
\(190\) −40.0000 −2.90191
\(191\) −23.0000 −1.66422 −0.832111 0.554609i \(-0.812868\pi\)
−0.832111 + 0.554609i \(0.812868\pi\)
\(192\) −8.00000 −0.577350
\(193\) −19.0000 −1.36765 −0.683825 0.729646i \(-0.739685\pi\)
−0.683825 + 0.729646i \(0.739685\pi\)
\(194\) −36.0000 −2.58465
\(195\) −8.00000 −0.572892
\(196\) 2.00000 0.142857
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) −10.0000 −0.710669
\(199\) 3.00000 0.212664 0.106332 0.994331i \(-0.466089\pi\)
0.106332 + 0.994331i \(0.466089\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 10.0000 0.703598
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) 20.0000 1.39686
\(206\) −38.0000 −2.64759
\(207\) −1.00000 −0.0695048
\(208\) 8.00000 0.554700
\(209\) 25.0000 1.72929
\(210\) −8.00000 −0.552052
\(211\) 13.0000 0.894957 0.447478 0.894295i \(-0.352322\pi\)
0.447478 + 0.894295i \(0.352322\pi\)
\(212\) 18.0000 1.23625
\(213\) 12.0000 0.822226
\(214\) 8.00000 0.546869
\(215\) 32.0000 2.18238
\(216\) 0 0
\(217\) −6.00000 −0.407307
\(218\) −24.0000 −1.62549
\(219\) 0 0
\(220\) −40.0000 −2.69680
\(221\) 0 0
\(222\) 12.0000 0.805387
\(223\) 10.0000 0.669650 0.334825 0.942280i \(-0.391323\pi\)
0.334825 + 0.942280i \(0.391323\pi\)
\(224\) 8.00000 0.534522
\(225\) 11.0000 0.733333
\(226\) −4.00000 −0.266076
\(227\) −18.0000 −1.19470 −0.597351 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) −10.0000 −0.662266
\(229\) −13.0000 −0.859064 −0.429532 0.903052i \(-0.641321\pi\)
−0.429532 + 0.903052i \(0.641321\pi\)
\(230\) −8.00000 −0.527504
\(231\) 5.00000 0.328976
\(232\) 0 0
\(233\) −12.0000 −0.786146 −0.393073 0.919507i \(-0.628588\pi\)
−0.393073 + 0.919507i \(0.628588\pi\)
\(234\) −4.00000 −0.261488
\(235\) −36.0000 −2.34838
\(236\) 18.0000 1.17170
\(237\) −10.0000 −0.649570
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) −16.0000 −1.03280
\(241\) 17.0000 1.09507 0.547533 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) 28.0000 1.79991
\(243\) 1.00000 0.0641500
\(244\) −10.0000 −0.640184
\(245\) 4.00000 0.255551
\(246\) 10.0000 0.637577
\(247\) 10.0000 0.636285
\(248\) 0 0
\(249\) −18.0000 −1.14070
\(250\) 48.0000 3.03579
\(251\) 10.0000 0.631194 0.315597 0.948893i \(-0.397795\pi\)
0.315597 + 0.948893i \(0.397795\pi\)
\(252\) −2.00000 −0.125988
\(253\) 5.00000 0.314347
\(254\) 18.0000 1.12942
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 17.0000 1.06043 0.530215 0.847863i \(-0.322111\pi\)
0.530215 + 0.847863i \(0.322111\pi\)
\(258\) 16.0000 0.996116
\(259\) −6.00000 −0.372822
\(260\) −16.0000 −0.992278
\(261\) −2.00000 −0.123797
\(262\) −10.0000 −0.617802
\(263\) 7.00000 0.431638 0.215819 0.976433i \(-0.430758\pi\)
0.215819 + 0.976433i \(0.430758\pi\)
\(264\) 0 0
\(265\) 36.0000 2.21146
\(266\) 10.0000 0.613139
\(267\) 10.0000 0.611990
\(268\) 8.00000 0.488678
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 8.00000 0.486864
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 0 0
\(273\) 2.00000 0.121046
\(274\) 18.0000 1.08742
\(275\) −55.0000 −3.31662
\(276\) −2.00000 −0.120386
\(277\) 7.00000 0.420589 0.210295 0.977638i \(-0.432558\pi\)
0.210295 + 0.977638i \(0.432558\pi\)
\(278\) 24.0000 1.43942
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) −18.0000 −1.07188
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 24.0000 1.42414
\(285\) −20.0000 −1.18470
\(286\) 20.0000 1.18262
\(287\) −5.00000 −0.295141
\(288\) −8.00000 −0.471405
\(289\) −17.0000 −1.00000
\(290\) −16.0000 −0.939552
\(291\) −18.0000 −1.05518
\(292\) 0 0
\(293\) −4.00000 −0.233682 −0.116841 0.993151i \(-0.537277\pi\)
−0.116841 + 0.993151i \(0.537277\pi\)
\(294\) 2.00000 0.116642
\(295\) 36.0000 2.09600
\(296\) 0 0
\(297\) −5.00000 −0.290129
\(298\) 6.00000 0.347571
\(299\) 2.00000 0.115663
\(300\) 22.0000 1.27017
\(301\) −8.00000 −0.461112
\(302\) −38.0000 −2.18665
\(303\) 5.00000 0.287242
\(304\) 20.0000 1.14708
\(305\) −20.0000 −1.14520
\(306\) 0 0
\(307\) −8.00000 −0.456584 −0.228292 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) 10.0000 0.569803
\(309\) −19.0000 −1.08087
\(310\) 48.0000 2.72622
\(311\) −15.0000 −0.850572 −0.425286 0.905059i \(-0.639826\pi\)
−0.425286 + 0.905059i \(0.639826\pi\)
\(312\) 0 0
\(313\) −25.0000 −1.41308 −0.706542 0.707671i \(-0.749746\pi\)
−0.706542 + 0.707671i \(0.749746\pi\)
\(314\) 14.0000 0.790066
\(315\) −4.00000 −0.225374
\(316\) −20.0000 −1.12509
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) 18.0000 1.00939
\(319\) 10.0000 0.559893
\(320\) −32.0000 −1.78885
\(321\) 4.00000 0.223258
\(322\) 2.00000 0.111456
\(323\) 0 0
\(324\) 2.00000 0.111111
\(325\) −22.0000 −1.22034
\(326\) 26.0000 1.44001
\(327\) −12.0000 −0.663602
\(328\) 0 0
\(329\) 9.00000 0.496186
\(330\) −40.0000 −2.20193
\(331\) 29.0000 1.59398 0.796992 0.603990i \(-0.206423\pi\)
0.796992 + 0.603990i \(0.206423\pi\)
\(332\) −36.0000 −1.97576
\(333\) 6.00000 0.328798
\(334\) 38.0000 2.07927
\(335\) 16.0000 0.874173
\(336\) 4.00000 0.218218
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) −18.0000 −0.979071
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) −30.0000 −1.62459
\(342\) −10.0000 −0.540738
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −4.00000 −0.215353
\(346\) 4.00000 0.215041
\(347\) 2.00000 0.107366 0.0536828 0.998558i \(-0.482904\pi\)
0.0536828 + 0.998558i \(0.482904\pi\)
\(348\) −4.00000 −0.214423
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) −22.0000 −1.17595
\(351\) −2.00000 −0.106752
\(352\) 40.0000 2.13201
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 18.0000 0.956689
\(355\) 48.0000 2.54758
\(356\) 20.0000 1.06000
\(357\) 0 0
\(358\) 16.0000 0.845626
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) −28.0000 −1.47165
\(363\) 14.0000 0.734809
\(364\) 4.00000 0.209657
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) 27.0000 1.40939 0.704694 0.709511i \(-0.251084\pi\)
0.704694 + 0.709511i \(0.251084\pi\)
\(368\) 4.00000 0.208514
\(369\) 5.00000 0.260290
\(370\) 48.0000 2.49540
\(371\) −9.00000 −0.467257
\(372\) 12.0000 0.622171
\(373\) −34.0000 −1.76045 −0.880227 0.474554i \(-0.842610\pi\)
−0.880227 + 0.474554i \(0.842610\pi\)
\(374\) 0 0
\(375\) 24.0000 1.23935
\(376\) 0 0
\(377\) 4.00000 0.206010
\(378\) −2.00000 −0.102869
\(379\) −30.0000 −1.54100 −0.770498 0.637442i \(-0.779993\pi\)
−0.770498 + 0.637442i \(0.779993\pi\)
\(380\) −40.0000 −2.05196
\(381\) 9.00000 0.461084
\(382\) −46.0000 −2.35356
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) 0 0
\(385\) 20.0000 1.01929
\(386\) −38.0000 −1.93415
\(387\) 8.00000 0.406663
\(388\) −36.0000 −1.82762
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) −16.0000 −0.810191
\(391\) 0 0
\(392\) 0 0
\(393\) −5.00000 −0.252217
\(394\) −4.00000 −0.201517
\(395\) −40.0000 −2.01262
\(396\) −10.0000 −0.502519
\(397\) 30.0000 1.50566 0.752828 0.658217i \(-0.228689\pi\)
0.752828 + 0.658217i \(0.228689\pi\)
\(398\) 6.00000 0.300753
\(399\) 5.00000 0.250313
\(400\) −44.0000 −2.20000
\(401\) 3.00000 0.149813 0.0749064 0.997191i \(-0.476134\pi\)
0.0749064 + 0.997191i \(0.476134\pi\)
\(402\) 8.00000 0.399004
\(403\) −12.0000 −0.597763
\(404\) 10.0000 0.497519
\(405\) 4.00000 0.198762
\(406\) 4.00000 0.198517
\(407\) −30.0000 −1.48704
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 40.0000 1.97546
\(411\) 9.00000 0.443937
\(412\) −38.0000 −1.87213
\(413\) −9.00000 −0.442861
\(414\) −2.00000 −0.0982946
\(415\) −72.0000 −3.53434
\(416\) 16.0000 0.784465
\(417\) 12.0000 0.587643
\(418\) 50.0000 2.44558
\(419\) −18.0000 −0.879358 −0.439679 0.898155i \(-0.644908\pi\)
−0.439679 + 0.898155i \(0.644908\pi\)
\(420\) −8.00000 −0.390360
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 26.0000 1.26566
\(423\) −9.00000 −0.437595
\(424\) 0 0
\(425\) 0 0
\(426\) 24.0000 1.16280
\(427\) 5.00000 0.241967
\(428\) 8.00000 0.386695
\(429\) 10.0000 0.482805
\(430\) 64.0000 3.08635
\(431\) 27.0000 1.30054 0.650272 0.759701i \(-0.274655\pi\)
0.650272 + 0.759701i \(0.274655\pi\)
\(432\) −4.00000 −0.192450
\(433\) −11.0000 −0.528626 −0.264313 0.964437i \(-0.585145\pi\)
−0.264313 + 0.964437i \(0.585145\pi\)
\(434\) −12.0000 −0.576018
\(435\) −8.00000 −0.383571
\(436\) −24.0000 −1.14939
\(437\) 5.00000 0.239182
\(438\) 0 0
\(439\) −36.0000 −1.71819 −0.859093 0.511819i \(-0.828972\pi\)
−0.859093 + 0.511819i \(0.828972\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 12.0000 0.569495
\(445\) 40.0000 1.89618
\(446\) 20.0000 0.947027
\(447\) 3.00000 0.141895
\(448\) 8.00000 0.377964
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 22.0000 1.03709
\(451\) −25.0000 −1.17720
\(452\) −4.00000 −0.188144
\(453\) −19.0000 −0.892698
\(454\) −36.0000 −1.68956
\(455\) 8.00000 0.375046
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −26.0000 −1.21490
\(459\) 0 0
\(460\) −8.00000 −0.373002
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 10.0000 0.465242
\(463\) −35.0000 −1.62659 −0.813294 0.581853i \(-0.802328\pi\)
−0.813294 + 0.581853i \(0.802328\pi\)
\(464\) 8.00000 0.371391
\(465\) 24.0000 1.11297
\(466\) −24.0000 −1.11178
\(467\) −6.00000 −0.277647 −0.138823 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) −4.00000 −0.184900
\(469\) −4.00000 −0.184703
\(470\) −72.0000 −3.32111
\(471\) 7.00000 0.322543
\(472\) 0 0
\(473\) −40.0000 −1.83920
\(474\) −20.0000 −0.918630
\(475\) −55.0000 −2.52357
\(476\) 0 0
\(477\) 9.00000 0.412082
\(478\) 24.0000 1.09773
\(479\) 32.0000 1.46212 0.731059 0.682315i \(-0.239027\pi\)
0.731059 + 0.682315i \(0.239027\pi\)
\(480\) −32.0000 −1.46059
\(481\) −12.0000 −0.547153
\(482\) 34.0000 1.54866
\(483\) 1.00000 0.0455016
\(484\) 28.0000 1.27273
\(485\) −72.0000 −3.26935
\(486\) 2.00000 0.0907218
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 0 0
\(489\) 13.0000 0.587880
\(490\) 8.00000 0.361403
\(491\) 14.0000 0.631811 0.315906 0.948791i \(-0.397692\pi\)
0.315906 + 0.948791i \(0.397692\pi\)
\(492\) 10.0000 0.450835
\(493\) 0 0
\(494\) 20.0000 0.899843
\(495\) −20.0000 −0.898933
\(496\) −24.0000 −1.07763
\(497\) −12.0000 −0.538274
\(498\) −36.0000 −1.61320
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 48.0000 2.14663
\(501\) 19.0000 0.848857
\(502\) 20.0000 0.892644
\(503\) −36.0000 −1.60516 −0.802580 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 0 0
\(505\) 20.0000 0.889988
\(506\) 10.0000 0.444554
\(507\) −9.00000 −0.399704
\(508\) 18.0000 0.798621
\(509\) −19.0000 −0.842160 −0.421080 0.907023i \(-0.638349\pi\)
−0.421080 + 0.907023i \(0.638349\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 32.0000 1.41421
\(513\) −5.00000 −0.220755
\(514\) 34.0000 1.49968
\(515\) −76.0000 −3.34896
\(516\) 16.0000 0.704361
\(517\) 45.0000 1.97910
\(518\) −12.0000 −0.527250
\(519\) 2.00000 0.0877903
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) −4.00000 −0.175075
\(523\) 19.0000 0.830812 0.415406 0.909636i \(-0.363640\pi\)
0.415406 + 0.909636i \(0.363640\pi\)
\(524\) −10.0000 −0.436852
\(525\) −11.0000 −0.480079
\(526\) 14.0000 0.610429
\(527\) 0 0
\(528\) 20.0000 0.870388
\(529\) 1.00000 0.0434783
\(530\) 72.0000 3.12748
\(531\) 9.00000 0.390567
\(532\) 10.0000 0.433555
\(533\) −10.0000 −0.433148
\(534\) 20.0000 0.865485
\(535\) 16.0000 0.691740
\(536\) 0 0
\(537\) 8.00000 0.345225
\(538\) −36.0000 −1.55207
\(539\) −5.00000 −0.215365
\(540\) 8.00000 0.344265
\(541\) 11.0000 0.472927 0.236463 0.971640i \(-0.424012\pi\)
0.236463 + 0.971640i \(0.424012\pi\)
\(542\) −16.0000 −0.687259
\(543\) −14.0000 −0.600798
\(544\) 0 0
\(545\) −48.0000 −2.05609
\(546\) 4.00000 0.171184
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 18.0000 0.768922
\(549\) −5.00000 −0.213395
\(550\) −110.000 −4.69042
\(551\) 10.0000 0.426014
\(552\) 0 0
\(553\) 10.0000 0.425243
\(554\) 14.0000 0.594803
\(555\) 24.0000 1.01874
\(556\) 24.0000 1.01783
\(557\) 46.0000 1.94908 0.974541 0.224208i \(-0.0719796\pi\)
0.974541 + 0.224208i \(0.0719796\pi\)
\(558\) 12.0000 0.508001
\(559\) −16.0000 −0.676728
\(560\) 16.0000 0.676123
\(561\) 0 0
\(562\) 4.00000 0.168730
\(563\) 40.0000 1.68580 0.842900 0.538071i \(-0.180847\pi\)
0.842900 + 0.538071i \(0.180847\pi\)
\(564\) −18.0000 −0.757937
\(565\) −8.00000 −0.336563
\(566\) −40.0000 −1.68133
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −17.0000 −0.712677 −0.356339 0.934357i \(-0.615975\pi\)
−0.356339 + 0.934357i \(0.615975\pi\)
\(570\) −40.0000 −1.67542
\(571\) 8.00000 0.334790 0.167395 0.985890i \(-0.446465\pi\)
0.167395 + 0.985890i \(0.446465\pi\)
\(572\) 20.0000 0.836242
\(573\) −23.0000 −0.960839
\(574\) −10.0000 −0.417392
\(575\) −11.0000 −0.458732
\(576\) −8.00000 −0.333333
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) −34.0000 −1.41421
\(579\) −19.0000 −0.789613
\(580\) −16.0000 −0.664364
\(581\) 18.0000 0.746766
\(582\) −36.0000 −1.49225
\(583\) −45.0000 −1.86371
\(584\) 0 0
\(585\) −8.00000 −0.330759
\(586\) −8.00000 −0.330477
\(587\) −17.0000 −0.701665 −0.350833 0.936438i \(-0.614101\pi\)
−0.350833 + 0.936438i \(0.614101\pi\)
\(588\) 2.00000 0.0824786
\(589\) −30.0000 −1.23613
\(590\) 72.0000 2.96419
\(591\) −2.00000 −0.0822690
\(592\) −24.0000 −0.986394
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) −10.0000 −0.410305
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 3.00000 0.122782
\(598\) 4.00000 0.163572
\(599\) 42.0000 1.71607 0.858037 0.513588i \(-0.171684\pi\)
0.858037 + 0.513588i \(0.171684\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) −16.0000 −0.652111
\(603\) 4.00000 0.162893
\(604\) −38.0000 −1.54620
\(605\) 56.0000 2.27672
\(606\) 10.0000 0.406222
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 40.0000 1.62221
\(609\) 2.00000 0.0810441
\(610\) −40.0000 −1.61955
\(611\) 18.0000 0.728202
\(612\) 0 0
\(613\) 34.0000 1.37325 0.686624 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) −16.0000 −0.645707
\(615\) 20.0000 0.806478
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) −38.0000 −1.52858
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 48.0000 1.92773
\(621\) −1.00000 −0.0401286
\(622\) −30.0000 −1.20289
\(623\) −10.0000 −0.400642
\(624\) 8.00000 0.320256
\(625\) 41.0000 1.64000
\(626\) −50.0000 −1.99840
\(627\) 25.0000 0.998404
\(628\) 14.0000 0.558661
\(629\) 0 0
\(630\) −8.00000 −0.318728
\(631\) −10.0000 −0.398094 −0.199047 0.979990i \(-0.563785\pi\)
−0.199047 + 0.979990i \(0.563785\pi\)
\(632\) 0 0
\(633\) 13.0000 0.516704
\(634\) 60.0000 2.38290
\(635\) 36.0000 1.42862
\(636\) 18.0000 0.713746
\(637\) −2.00000 −0.0792429
\(638\) 20.0000 0.791808
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) −9.00000 −0.355479 −0.177739 0.984078i \(-0.556878\pi\)
−0.177739 + 0.984078i \(0.556878\pi\)
\(642\) 8.00000 0.315735
\(643\) −31.0000 −1.22252 −0.611260 0.791430i \(-0.709337\pi\)
−0.611260 + 0.791430i \(0.709337\pi\)
\(644\) 2.00000 0.0788110
\(645\) 32.0000 1.26000
\(646\) 0 0
\(647\) −28.0000 −1.10079 −0.550397 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) 0 0
\(649\) −45.0000 −1.76640
\(650\) −44.0000 −1.72582
\(651\) −6.00000 −0.235159
\(652\) 26.0000 1.01824
\(653\) 24.0000 0.939193 0.469596 0.882881i \(-0.344399\pi\)
0.469596 + 0.882881i \(0.344399\pi\)
\(654\) −24.0000 −0.938474
\(655\) −20.0000 −0.781465
\(656\) −20.0000 −0.780869
\(657\) 0 0
\(658\) 18.0000 0.701713
\(659\) 24.0000 0.934907 0.467454 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) −40.0000 −1.55700
\(661\) −5.00000 −0.194477 −0.0972387 0.995261i \(-0.531001\pi\)
−0.0972387 + 0.995261i \(0.531001\pi\)
\(662\) 58.0000 2.25423
\(663\) 0 0
\(664\) 0 0
\(665\) 20.0000 0.775567
\(666\) 12.0000 0.464991
\(667\) 2.00000 0.0774403
\(668\) 38.0000 1.47026
\(669\) 10.0000 0.386622
\(670\) 32.0000 1.23627
\(671\) 25.0000 0.965114
\(672\) 8.00000 0.308607
\(673\) 49.0000 1.88881 0.944406 0.328783i \(-0.106638\pi\)
0.944406 + 0.328783i \(0.106638\pi\)
\(674\) 44.0000 1.69482
\(675\) 11.0000 0.423390
\(676\) −18.0000 −0.692308
\(677\) 36.0000 1.38359 0.691796 0.722093i \(-0.256820\pi\)
0.691796 + 0.722093i \(0.256820\pi\)
\(678\) −4.00000 −0.153619
\(679\) 18.0000 0.690777
\(680\) 0 0
\(681\) −18.0000 −0.689761
\(682\) −60.0000 −2.29752
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) −10.0000 −0.382360
\(685\) 36.0000 1.37549
\(686\) −2.00000 −0.0763604
\(687\) −13.0000 −0.495981
\(688\) −32.0000 −1.21999
\(689\) −18.0000 −0.685745
\(690\) −8.00000 −0.304555
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) 4.00000 0.152057
\(693\) 5.00000 0.189934
\(694\) 4.00000 0.151838
\(695\) 48.0000 1.82074
\(696\) 0 0
\(697\) 0 0
\(698\) −4.00000 −0.151402
\(699\) −12.0000 −0.453882
\(700\) −22.0000 −0.831522
\(701\) −15.0000 −0.566542 −0.283271 0.959040i \(-0.591420\pi\)
−0.283271 + 0.959040i \(0.591420\pi\)
\(702\) −4.00000 −0.150970
\(703\) −30.0000 −1.13147
\(704\) 40.0000 1.50756
\(705\) −36.0000 −1.35584
\(706\) −28.0000 −1.05379
\(707\) −5.00000 −0.188044
\(708\) 18.0000 0.676481
\(709\) −6.00000 −0.225335 −0.112667 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) 96.0000 3.60282
\(711\) −10.0000 −0.375029
\(712\) 0 0
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) 40.0000 1.49592
\(716\) 16.0000 0.597948
\(717\) 12.0000 0.448148
\(718\) 64.0000 2.38846
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) −16.0000 −0.596285
\(721\) 19.0000 0.707597
\(722\) 12.0000 0.446594
\(723\) 17.0000 0.632237
\(724\) −28.0000 −1.04061
\(725\) −22.0000 −0.817059
\(726\) 28.0000 1.03918
\(727\) 41.0000 1.52061 0.760303 0.649569i \(-0.225051\pi\)
0.760303 + 0.649569i \(0.225051\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) −10.0000 −0.369611
\(733\) −6.00000 −0.221615 −0.110808 0.993842i \(-0.535344\pi\)
−0.110808 + 0.993842i \(0.535344\pi\)
\(734\) 54.0000 1.99318
\(735\) 4.00000 0.147542
\(736\) 8.00000 0.294884
\(737\) −20.0000 −0.736709
\(738\) 10.0000 0.368105
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 48.0000 1.76452
\(741\) 10.0000 0.367359
\(742\) −18.0000 −0.660801
\(743\) 15.0000 0.550297 0.275148 0.961402i \(-0.411273\pi\)
0.275148 + 0.961402i \(0.411273\pi\)
\(744\) 0 0
\(745\) 12.0000 0.439646
\(746\) −68.0000 −2.48966
\(747\) −18.0000 −0.658586
\(748\) 0 0
\(749\) −4.00000 −0.146157
\(750\) 48.0000 1.75271
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 36.0000 1.31278
\(753\) 10.0000 0.364420
\(754\) 8.00000 0.291343
\(755\) −76.0000 −2.76592
\(756\) −2.00000 −0.0727393
\(757\) 8.00000 0.290765 0.145382 0.989376i \(-0.453559\pi\)
0.145382 + 0.989376i \(0.453559\pi\)
\(758\) −60.0000 −2.17930
\(759\) 5.00000 0.181489
\(760\) 0 0
\(761\) −21.0000 −0.761249 −0.380625 0.924730i \(-0.624291\pi\)
−0.380625 + 0.924730i \(0.624291\pi\)
\(762\) 18.0000 0.652071
\(763\) 12.0000 0.434429
\(764\) −46.0000 −1.66422
\(765\) 0 0
\(766\) 12.0000 0.433578
\(767\) −18.0000 −0.649942
\(768\) 16.0000 0.577350
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 40.0000 1.44150
\(771\) 17.0000 0.612240
\(772\) −38.0000 −1.36765
\(773\) 20.0000 0.719350 0.359675 0.933078i \(-0.382888\pi\)
0.359675 + 0.933078i \(0.382888\pi\)
\(774\) 16.0000 0.575108
\(775\) 66.0000 2.37079
\(776\) 0 0
\(777\) −6.00000 −0.215249
\(778\) 12.0000 0.430221
\(779\) −25.0000 −0.895718
\(780\) −16.0000 −0.572892
\(781\) −60.0000 −2.14697
\(782\) 0 0
\(783\) −2.00000 −0.0714742
\(784\) −4.00000 −0.142857
\(785\) 28.0000 0.999363
\(786\) −10.0000 −0.356688
\(787\) −23.0000 −0.819861 −0.409931 0.912117i \(-0.634447\pi\)
−0.409931 + 0.912117i \(0.634447\pi\)
\(788\) −4.00000 −0.142494
\(789\) 7.00000 0.249207
\(790\) −80.0000 −2.84627
\(791\) 2.00000 0.0711118
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) 60.0000 2.12932
\(795\) 36.0000 1.27679
\(796\) 6.00000 0.212664
\(797\) 46.0000 1.62940 0.814702 0.579880i \(-0.196901\pi\)
0.814702 + 0.579880i \(0.196901\pi\)
\(798\) 10.0000 0.353996
\(799\) 0 0
\(800\) −88.0000 −3.11127
\(801\) 10.0000 0.353333
\(802\) 6.00000 0.211867
\(803\) 0 0
\(804\) 8.00000 0.282138
\(805\) 4.00000 0.140981
\(806\) −24.0000 −0.845364
\(807\) −18.0000 −0.633630
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 8.00000 0.281091
\(811\) 2.00000 0.0702295 0.0351147 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) 4.00000 0.140372
\(813\) −8.00000 −0.280572
\(814\) −60.0000 −2.10300
\(815\) 52.0000 1.82148
\(816\) 0 0
\(817\) −40.0000 −1.39942
\(818\) 28.0000 0.978997
\(819\) 2.00000 0.0698857
\(820\) 40.0000 1.39686
\(821\) −4.00000 −0.139601 −0.0698005 0.997561i \(-0.522236\pi\)
−0.0698005 + 0.997561i \(0.522236\pi\)
\(822\) 18.0000 0.627822
\(823\) 27.0000 0.941161 0.470580 0.882357i \(-0.344045\pi\)
0.470580 + 0.882357i \(0.344045\pi\)
\(824\) 0 0
\(825\) −55.0000 −1.91485
\(826\) −18.0000 −0.626300
\(827\) −21.0000 −0.730242 −0.365121 0.930960i \(-0.618972\pi\)
−0.365121 + 0.930960i \(0.618972\pi\)
\(828\) −2.00000 −0.0695048
\(829\) 26.0000 0.903017 0.451509 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) −144.000 −4.99831
\(831\) 7.00000 0.242827
\(832\) 16.0000 0.554700
\(833\) 0 0
\(834\) 24.0000 0.831052
\(835\) 76.0000 2.63009
\(836\) 50.0000 1.72929
\(837\) 6.00000 0.207390
\(838\) −36.0000 −1.24360
\(839\) 46.0000 1.58810 0.794048 0.607855i \(-0.207970\pi\)
0.794048 + 0.607855i \(0.207970\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 16.0000 0.551396
\(843\) 2.00000 0.0688837
\(844\) 26.0000 0.894957
\(845\) −36.0000 −1.23844
\(846\) −18.0000 −0.618853
\(847\) −14.0000 −0.481046
\(848\) −36.0000 −1.23625
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) −6.00000 −0.205677
\(852\) 24.0000 0.822226
\(853\) 8.00000 0.273915 0.136957 0.990577i \(-0.456268\pi\)
0.136957 + 0.990577i \(0.456268\pi\)
\(854\) 10.0000 0.342193
\(855\) −20.0000 −0.683986
\(856\) 0 0
\(857\) 1.00000 0.0341593 0.0170797 0.999854i \(-0.494563\pi\)
0.0170797 + 0.999854i \(0.494563\pi\)
\(858\) 20.0000 0.682789
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) 64.0000 2.18238
\(861\) −5.00000 −0.170400
\(862\) 54.0000 1.83925
\(863\) −30.0000 −1.02121 −0.510606 0.859815i \(-0.670579\pi\)
−0.510606 + 0.859815i \(0.670579\pi\)
\(864\) −8.00000 −0.272166
\(865\) 8.00000 0.272008
\(866\) −22.0000 −0.747590
\(867\) −17.0000 −0.577350
\(868\) −12.0000 −0.407307
\(869\) 50.0000 1.69613
\(870\) −16.0000 −0.542451
\(871\) −8.00000 −0.271070
\(872\) 0 0
\(873\) −18.0000 −0.609208
\(874\) 10.0000 0.338255
\(875\) −24.0000 −0.811348
\(876\) 0 0
\(877\) −17.0000 −0.574049 −0.287025 0.957923i \(-0.592666\pi\)
−0.287025 + 0.957923i \(0.592666\pi\)
\(878\) −72.0000 −2.42988
\(879\) −4.00000 −0.134917
\(880\) 80.0000 2.69680
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) 2.00000 0.0673435
\(883\) 32.0000 1.07689 0.538443 0.842662i \(-0.319013\pi\)
0.538443 + 0.842662i \(0.319013\pi\)
\(884\) 0 0
\(885\) 36.0000 1.21013
\(886\) −72.0000 −2.41889
\(887\) −16.0000 −0.537227 −0.268614 0.963248i \(-0.586566\pi\)
−0.268614 + 0.963248i \(0.586566\pi\)
\(888\) 0 0
\(889\) −9.00000 −0.301850
\(890\) 80.0000 2.68161
\(891\) −5.00000 −0.167506
\(892\) 20.0000 0.669650
\(893\) 45.0000 1.50587
\(894\) 6.00000 0.200670
\(895\) 32.0000 1.06964
\(896\) 0 0
\(897\) 2.00000 0.0667781
\(898\) 12.0000 0.400445
\(899\) −12.0000 −0.400222
\(900\) 22.0000 0.733333
\(901\) 0 0
\(902\) −50.0000 −1.66482
\(903\) −8.00000 −0.266223
\(904\) 0 0
\(905\) −56.0000 −1.86150
\(906\) −38.0000 −1.26247
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) −36.0000 −1.19470
\(909\) 5.00000 0.165840
\(910\) 16.0000 0.530395
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 20.0000 0.662266
\(913\) 90.0000 2.97857
\(914\) 20.0000 0.661541
\(915\) −20.0000 −0.661180
\(916\) −26.0000 −0.859064
\(917\) 5.00000 0.165115
\(918\) 0 0
\(919\) 2.00000 0.0659739 0.0329870 0.999456i \(-0.489498\pi\)
0.0329870 + 0.999456i \(0.489498\pi\)
\(920\) 0 0
\(921\) −8.00000 −0.263609
\(922\) 4.00000 0.131733
\(923\) −24.0000 −0.789970
\(924\) 10.0000 0.328976
\(925\) 66.0000 2.17007
\(926\) −70.0000 −2.30034
\(927\) −19.0000 −0.624042
\(928\) 16.0000 0.525226
\(929\) 26.0000 0.853032 0.426516 0.904480i \(-0.359741\pi\)
0.426516 + 0.904480i \(0.359741\pi\)
\(930\) 48.0000 1.57398
\(931\) −5.00000 −0.163868
\(932\) −24.0000 −0.786146
\(933\) −15.0000 −0.491078
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) 0 0
\(937\) −27.0000 −0.882052 −0.441026 0.897494i \(-0.645385\pi\)
−0.441026 + 0.897494i \(0.645385\pi\)
\(938\) −8.00000 −0.261209
\(939\) −25.0000 −0.815844
\(940\) −72.0000 −2.34838
\(941\) 20.0000 0.651981 0.325991 0.945373i \(-0.394302\pi\)
0.325991 + 0.945373i \(0.394302\pi\)
\(942\) 14.0000 0.456145
\(943\) −5.00000 −0.162822
\(944\) −36.0000 −1.17170
\(945\) −4.00000 −0.130120
\(946\) −80.0000 −2.60102
\(947\) 32.0000 1.03986 0.519930 0.854209i \(-0.325958\pi\)
0.519930 + 0.854209i \(0.325958\pi\)
\(948\) −20.0000 −0.649570
\(949\) 0 0
\(950\) −110.000 −3.56887
\(951\) 30.0000 0.972817
\(952\) 0 0
\(953\) −55.0000 −1.78162 −0.890812 0.454371i \(-0.849864\pi\)
−0.890812 + 0.454371i \(0.849864\pi\)
\(954\) 18.0000 0.582772
\(955\) −92.0000 −2.97705
\(956\) 24.0000 0.776215
\(957\) 10.0000 0.323254
\(958\) 64.0000 2.06775
\(959\) −9.00000 −0.290625
\(960\) −32.0000 −1.03280
\(961\) 5.00000 0.161290
\(962\) −24.0000 −0.773791
\(963\) 4.00000 0.128898
\(964\) 34.0000 1.09507
\(965\) −76.0000 −2.44653
\(966\) 2.00000 0.0643489
\(967\) 28.0000 0.900419 0.450210 0.892923i \(-0.351349\pi\)
0.450210 + 0.892923i \(0.351349\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −144.000 −4.62356
\(971\) −18.0000 −0.577647 −0.288824 0.957382i \(-0.593264\pi\)
−0.288824 + 0.957382i \(0.593264\pi\)
\(972\) 2.00000 0.0641500
\(973\) −12.0000 −0.384702
\(974\) −40.0000 −1.28168
\(975\) −22.0000 −0.704564
\(976\) 20.0000 0.640184
\(977\) −3.00000 −0.0959785 −0.0479893 0.998848i \(-0.515281\pi\)
−0.0479893 + 0.998848i \(0.515281\pi\)
\(978\) 26.0000 0.831388
\(979\) −50.0000 −1.59801
\(980\) 8.00000 0.255551
\(981\) −12.0000 −0.383131
\(982\) 28.0000 0.893516
\(983\) 42.0000 1.33959 0.669796 0.742545i \(-0.266382\pi\)
0.669796 + 0.742545i \(0.266382\pi\)
\(984\) 0 0
\(985\) −8.00000 −0.254901
\(986\) 0 0
\(987\) 9.00000 0.286473
\(988\) 20.0000 0.636285
\(989\) −8.00000 −0.254385
\(990\) −40.0000 −1.27128
\(991\) 5.00000 0.158830 0.0794151 0.996842i \(-0.474695\pi\)
0.0794151 + 0.996842i \(0.474695\pi\)
\(992\) −48.0000 −1.52400
\(993\) 29.0000 0.920287
\(994\) −24.0000 −0.761234
\(995\) 12.0000 0.380426
\(996\) −36.0000 −1.14070
\(997\) −56.0000 −1.77354 −0.886769 0.462213i \(-0.847056\pi\)
−0.886769 + 0.462213i \(0.847056\pi\)
\(998\) −40.0000 −1.26618
\(999\) 6.00000 0.189832
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 483.2.a.b.1.1 1
3.2 odd 2 1449.2.a.a.1.1 1
4.3 odd 2 7728.2.a.l.1.1 1
7.6 odd 2 3381.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.b.1.1 1 1.1 even 1 trivial
1449.2.a.a.1.1 1 3.2 odd 2
3381.2.a.l.1.1 1 7.6 odd 2
7728.2.a.l.1.1 1 4.3 odd 2